Probability from Venn Diagrams

A Venn diagram is a diagrammatic representation of sets where each set is represented by a closed curve.
The total space within the Venn diagram represents the universal set which includes all of the possible outcomes.
Each circle represents a subset of the universal set.
The overlap between circles represents the intersection of the sets, which is the set of elements common to both sets.
If the circles do not overlap, this means the sets have no common elements and are mutually exclusive.

In a Venn diagram, the probability of an event is given by the number of favourable outcomes (elements in the set of interest) divided by the total number of outcomes (elements in the universal set).
The probability of the union of two events (A ∪ B) is found by adding the probabilities of events A and B, and subtracting the probability of their intersection (A ∩ B). This is the principle of addition of probabilities.
The probability of the intersection of two events (A ∩ B) is given by the number of elements common to both A and B divided by the total number of elements in the universal set.
In a Venn diagram, independent events do not affect each other’s probability, while the occurrence of one dependent event changes the probability of the other event.
The complement of an event A, denoted as A’, is the set of outcomes in the universal set that are not in A. The probability of the complement of an event, P(A’), is equal to 1 - P(A).

Venn diagrams are helpful in visualising relationships between events and their probabilities.
When reading a Venn diagram, start by identifying the universal set and then identify the subsets represented by the circles. Understand which elements belong to which sets and where intersections and unions occur.
Problem-solving with Venn diagrams often involves working backwards, using given probabilities to find the number of elements in different sections of the diagram.

Venn diagrams can model real-world situations in fields like business, science, politics or sports, and help in making predictions and decisions.
Misinterpretation of Venn diagrams can lead to miscalculations in probability, so always verify the given information and your calculations carefully.
Practise applying Venn diagrams in different contexts to improve your understanding of this tool and its use in statistics.